I have an external hard drive (Lacie big disc extreme). The power supply melted and my warranty is void. I need the data on this disc but do not want to fork over the $50 to replace the power supply right now.

I have opened the case and found two 500GB drives. I can only assume they are in RAID 0.

Can I safely access these drives by plugging them directly into my desktop PC?
Should I replicate the RAID 0 on my desktop?

Don't despair! You're not totally stuffed! THERE ARE RECOVERY OPTIONS! If only the PSU is dead, there is a piece of software I've used called Zero Assumption Recovery (and it's not the only software that can do this, but its the one I've got experience with).